Since $ P(\mu_k)=-\alpha_k Q^{\prime}(\mu_k)\not=0$, the polynomials
$ P$ and $ Q$ have no common roots. Thus the ratio in the right hand
side of (1.4) is irreducible. The Eq. (1.1) is equivalent
to the equation

$ \begin{equation*} \label{EqEq} P(z)-tQ(z)=0. \end{equation*}$

(1.8)

Since the polynomial $ P(z)-tQ(z)$ is of degree $ n+1$, the latter
equation has $ n+1$ roots for each $ t\in\mathbb{C}$.

Since $ R^{\prime}(x)> 0$ for
$ x\in\mathbb{R},\,x\not=\mu_1,\ldots,\mu_n$, each of the functions
$ \nu_k(t),k=0,1,\ldots,n$, can be continued as a single valued
holomorphic function to some neighborhood of $ \mathbb{R}$. However
the functions $ \nu_k(t)$ can not be continued as single-valued
analytic functions to the whole complex $ t$-plane. According to
(1.4),

The polynomial $ P^{\prime}Q-Q^{\prime}P$ is of degree $ 2n$ and is
strictly positive on the real axis. Therefore this polynomial has
$ n$ roots $ \zeta_1,\ldots,\zeta_n$ in the upper half-plane
$ \text{Im}(z)> 0$ and $ n$ roots
$ \overline{\zeta_1},\ldots,\overline{\zeta_n}$ in the lower
half-plane $ \text{Im}(z)< 0$. (Not all roots $ \zeta_1,\ldots,\zeta_n$
must be distinct.) The points $ \zeta_1,\ldots,\zeta_n$ and
$ \overline{\zeta_1},\ldots,\overline{\zeta_n}$ are the critical
points of the function $ R$:
$ R^{\prime}(\zeta_k)=0,\,R^{\prime}(\overline{\zeta_k})=0,\ 1\leq k\leq n.$ The critical values
$ t_k=R(\zeta_k),\,\overline{t_k}=R(\overline{\zeta_k}),\ 1\leq k\leq n,$ of the function $ R$ are the ramification points of the function
$ \nu(t)$:

$ \begin{equation*} \label{RoF} R(\nu(t))=t \end{equation*}$

(1.13)

(Even if the critical points $ \zeta^{\prime}$ and
$ \zeta^{\prime\prime}$ of $ R$ are distinct, the critical values
$ R(\zeta^{\prime})$ and $ R(\zeta^{\prime\prime})$ may coincide.) We
denote the set of critical values of the function $ R$ by
$ \mathcal{V}$:

Let $ G$ be an arbitrary simply connected domain in the $ t$-plane
which does not intersect the set $ \mathcal{V}$. Then the roots of
Eq. (1.1) are pairwise distinct for each $ t\in{}G$. We can
enumerate these roots, say $ \nu_0(t),\nu_1(t),\,\ldots\,\nu_n(t)$,
such that all functions $ \nu_k(t)$ are holomorphic in $ G$.

does not intersect the set $ \mathcal{V}$. So $ n+1$ single valued
holomorphic branches of the function $ \nu(t)$, (1.13), are
defined in the strip $ S_h$. We choose such enumeration of
these branches which agrees with the enumeration (1.10)
on $ \mathbb{R}$.

From (1.6) and (1.2) it follows that the polynomial $ P$
is representable in the form

The polynomial pencil $ P(z)-tQ(z)$ is hyperbolic: for each
real $ t$, all roots of the Eq. (1.8) are real.

Using (1.17), we represent the polynomial $ P(z)-tQ(z)$ as the
characteristic polynomial $ \det(zI-(A+tB))$ of some matrix pencil,
where $ A$ and $ B$ are self-adjoint $ (n+1)\times(n+1)$ matrices,
$ \text{rank}\,B=1$. We present these matrices explicitly.

Theorem 2.2.

Let $ R$ be a function of the form (1.2), where
$ \mu_1,\mu_2,\ldots,\mu_n$ are pairwise distinct real numbers and
$ \alpha_1,\alpha_2,\ldots,\alpha_n$ are positive numbers. Let $ Q$
and $ P$ be the polynomials related to the the function $ R$ by the
equalities (1.5) and (1.17).

Then the pencil of polynomials $ P(z)-tQ(z)$ is representable as
the characteristic polynomial of the matrix pencil $ A+tB$, i.e.
the equality (2.5) holds for every $ z\in\mathbb{C}, t\in\mathbb{C}$, where $ B$ is the matrix with the entries
(2.2), and the entries of the matrix $ A$ are defined by by
(2.1) with

Proof.

We refer to Corollary 2.3. If $ \nu$ is an eigenvalue of some
square matrix $ M$, then $ h(\nu)$ is an eigenvalue of the matrix
$ h(M)$. In (2.8), we interpret the trace of the matrix
$ h(A+tB)$ as its spectral trace, that is as the sum of all
its eigenvalues.
$ \square$

are non-negative for every choice of complex numbers
$ \zeta_1,\zeta_2,\,\ldots\,,\zeta_N$ and for every choice of real
numbers $ t_1,t_2,\,\ldots\,,t_N$ assuming that all sums
$ t_r+t_s$ are within the interval $ (a,b)$.

P 4. Let $ \lbrace f_{n}(t)\rbrace_{1\leq n< \infty}$ be a sequence of functions from the class $ W_{a,b}$. We assume that for each $ t\in(a,b)$ there exists a limit
$ f(t)=\lim_{n\to\infty}f_{n}(t)$, and that $ f(t)< \infty\ \forall t\in(a,b)$.
Then $ f(t)\in{}W_{a,b}$.

From the functional equation for the exponential function it follows
that for each real number $ u$, for every choice of real numbers
$ t_1,t_2,\ldots,$ $ t_{N}$ and complex numbers $ \zeta_1$, $ \zeta_2, \ldots, \zeta_{N}$, the equality holds

is non-negative because this sum is the trace of a non-negative
matrix. The measure $ \sigma$ in the integral representation
(3.3) of the function $ \varphi$, (4.1), is an atomic
measure supported on the spectrum of the matrix $ V$.

In the general case, if the matrices $ U$ and $ V$ do not commute, the
BMV conjecture remained an open question for longer than 40 years.
In 2011, Herbert Stahl proved the BMV conjecture.

Theorem 4.1.

(H. Stahl) Let $ U$ and $ V$ be Hermitian matrices.

Then the function $ \varphi(t)$ defined by (4.1) belongs to the class $ W_{-\infty,\infty}$ of functions exponentially convex on $ (-\infty,\infty)$.

The first arXiv version of Stahl’s Theorem appeared in [Stahl2011],
the latest arXiv version—in [Stahl2012], the journal
publication—in [Stahl2013].

The proof of Herbert Stahl is based on ingenious considerations
related to Riemann surfaces of algebraic functions. In [Eremenko2015], a
simplified version of the Herbert Stahl proof is presented.

We present a toy version of Theorem 4.1 which is enough for
our goal.

Theorem 4.2.

Let $ U$ and $ V$ be Hermitian matrices. We assume moreover that

1.

All off-diagonal entries of the matrix
$ U$
are non-negative.

2.

The matrix
$ V$
is diagonal.

Then the function $ \varphi(t)$ defined by (4.1) belongs to the class $ W_{-\infty,\infty}$.

Proof.

For $ \rho\geq 0$, let $ U_{\rho}=U+\rho{}I$, where $ I$ is the
identity matrix. If $ \rho$ is large enough, then all entries of
the matrix $ U_{\rho}$ are non-negative. Let us choose and fix such
$ \rho$. It is clear that

where $ v_1,v_2,\ldots,v_m$ are real numbers. The exponentials
$ e^{tv_j/m}$ are functions of $ t$ from the class
$ W_{-\infty,\infty}$. Each entry of the matrix
$ e^{U_{\rho}/m}\,e^{tV/m}$ is a linear combination of these
exponentials with non-negative coefficients. According to the
properties P1 and P2 of the class $ W_{-\infty,\infty}$, the entries
of the matrix $ e^{U_{\rho}/m}\,e^{tV/m}$ are functions of the class
$ W_{-\infty,\infty}$. Each entry of the matrix
$ (e^{U_{\rho}/m}\,e^{tV/m})^m$ is a sum of products of some entries
of the matrix $ e^{U_{\rho}/m}\,e^{tV/m}$. According to the
properties P2 and P3 of the class $ W_{-\infty,\infty}$, the entries
of the matrix $ (e^{U_{\rho}/m}\,e^{tV/m})^m$ are functions of $ t$
belonging to the class $ W_{-\infty,\infty}$. From the limiting
relation (4.3) and from the property P4 of the class
$ W_{-\infty,\infty}$ it follows that all entries of the matrix
$ e^{U_{\rho}+tV}$ are functions of $ t$ belonging to the class
$ W_{-\infty,\infty}$. From (4.2) it follows that all entries
of the matrix $ e^{U+tV}$ belong to the class $ W_{-\infty,\infty}$.
All the more, the function $ \varphi(t)=\text{trace}\,\{e^{U+tV}\}$,
which is the sum of diagonal entries of the matrix $ e^{U+tV}$,
belongs to the class $ W_{-\infty,\infty}$. $ \square$

Lemma 5.1.

Let $ R$ be the rational function of the form (1.2),
$ \nu_{0}(t),\nu_1(t),\ldots,$ $ \nu_n(t)$ be the roots of the Eq.
(1.1). Let $ A$ and $ B$ be the matrices (2.1),
(2.6), (2.2) which appear in the determinant
representation (2.5) of the matrix pencil $ P(z)-tQ(z)$.

Remark 5.4.

Familiarizing himself with our proof of Theorem 5.2, Alexey
Kuznetsov (http://www.math.yorku.ca/ãkuznets/) gave a new
proof of a somewhat weakened version of this theorem. His proof is
based on the theory of stochastic Lévy processes.